Fourier transform - Revision history

KevinYager: /* See Also */

2015-07-19T15:12:02Z

‎See Also

KevinYager: /* Scattering */

2015-01-14T21:52:38Z

‎Scattering

KevinYager: /* Scattering */

2015-01-14T21:38:40Z

‎Scattering

KevinYager: /* Phase problem */

2014-12-16T23:08:37Z

‎Phase problem

KevinYager at 17:11, 30 September 2014

2014-09-30T17:11:48Z

KevinYager at 16:31, 30 September 2014

2014-09-30T16:31:43Z

KevinYager at 13:35, 25 August 2014

2014-08-25T13:35:53Z

KevinYager: /* Phase problem */

2014-06-05T21:37:07Z

‎Phase problem

KevinYager: /* Scattering */

2014-06-05T21:36:16Z

‎Scattering

KevinYager: /* Scattering */

2014-06-05T21:34:03Z

‎Scattering

← Older revision		Revision as of 15:12, 19 July 2015
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	==See Also==		==See Also==
	* [http://en.wikipedia.org/wiki/Fourier_transform Wikipedia: Fourier transform]		* [http://en.wikipedia.org/wiki/Fourier_transform Wikipedia: Fourier transform]
		+	* Geometric tutorial:
		+	** [http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/ An Interactive Guide To The Fourier Transform]
		+	** [https://www.youtube.com/watch?v=LznjC4Lo7lE Fourier Series Animation using Circles]

@@ Line 17: / Line 17: @@
 \end{alignat}
 </math>
-Where the observed [[scattering intensity]] (''I'') in the 3D reciprocal space ('''q''') is given by an ensemble average of the intensity for all (''N'') scattering entities probed by the beam. The wave-matter interaction is given by inner term, which coherently sums (interferes) the complex scattering contributions (<math>\rho_n</math> denotes the scattering power) of the ''N'' entities; the exponential term represents a plane wave (incident radiation). In integral form (for a continuous function of the scattering density <math>\rho(\mathbf{r})</math>), we can write an integral over all of real-space:
+Where the observed [[scattering intensity]] (''I'') in the 3D reciprocal-space ('''q''') is given by an ensemble average of the intensity for all (''N'') scattering entities probed by the beam. The wave-matter interaction is given by inner term, which coherently sums (interferes) the complex scattering contributions (<math>\rho_n</math> denotes the scattering power) of the ''N'' entities; the exponential term represents a plane wave (incident radiation). In integral form (for a continuous function of the scattering density <math>\rho(\mathbf{r})</math>), we can write an integral over all of real-space:
 :<math>
 \begin{alignat}{2}

@@ Line 17: / Line 17: @@
 \end{alignat}
 </math>
-Where the observed scattering intensity (''I'') in the 3D reciprocal space ('''q''') is given by an ensemble average of the intensity for all (''N'') scattering entities probed by the beam. The wave-matter interaction is given by inner term, which coherently sums (interferes) the complex scattering contributions (<math>\rho_n</math> denotes the scattering power) of the ''N'' entities; the exponential term represents a plane wave (incident radiation). In integral form (for a continuous function of the scattering density <math>\rho(\mathbf{r})</math>), we can write an integral over all of real-space:
+Where the observed [[scattering intensity]] (''I'') in the 3D reciprocal space ('''q''') is given by an ensemble average of the intensity for all (''N'') scattering entities probed by the beam. The wave-matter interaction is given by inner term, which coherently sums (interferes) the complex scattering contributions (<math>\rho_n</math> denotes the scattering power) of the ''N'' entities; the exponential term represents a plane wave (incident radiation). In integral form (for a continuous function of the scattering density <math>\rho(\mathbf{r})</math>), we can write an integral over all of real-space:
 :<math>
 \begin{alignat}{2}

@@ Line 35: / Line 35: @@
 ==Phase problem==
-Note that experimentally, we can only measure the squared amplitude, ''I''('''q'''); we can never directly measure the ''true'' reciprocal-space, given by ''F''('''q'''). This is known as the ''phase-problem'': our detector only measures the intensity of the scattered radiation, and not the phase of the scattered waves. As such, the scattering dataset is incomplete. If we had both the amplitude and phase of reciprocal-space, we could simply invert the data (using an inverse Fourier transform), and thereby recover the realspace structure of the sample. Lacking the phase information, we cannot do this. Instead, scattering data must be fit to candidate models, with concordant models being deemed 'more likely' to represent the sample. However, interpreting scattering data is a formally ill-prosed problem: there are multiple possible realspace models that will fit a particular experimental dataset. Abstractly, it is impossible to know which model is the right one. In practice, one can use auxilliary knowledge, such as other measurements (AFM, electron microscopy) or physical constraints, in order to exclude some candidate models, and thereby identify the correct sample structure.
+Note that experimentally, we can only measure the squared amplitude, ''I''('''q'''); we can never directly measure the ''true'' reciprocal-space, given by ''F''('''q'''). This is known as the ''phase-problem'': our [[detector]] only measures the intensity of the scattered radiation, and not the phase of the scattered waves. As such, the scattering dataset is incomplete. If we had both the amplitude and phase of reciprocal-space, we could simply invert the data (using an inverse Fourier transform), and thereby recover the realspace structure of the sample. Lacking the phase information, we cannot do this. Instead, scattering data must be fit to candidate models, with concordant models being deemed 'more likely' to represent the sample. However, interpreting scattering data is a formally ill-prosed problem: there are multiple possible realspace models that will fit a particular experimental dataset. Abstractly, it is impossible to know which model is the right one. In practice, one can use auxilliary knowledge, such as other measurements (AFM, electron microscopy) or physical constraints, in order to exclude some candidate models, and thereby identify the correct sample structure.
 There are many interesting research thrusts aimed at 'solving' the phase problem. For instance, one can conduct repeated measurements on the same (or highly similar) sample using different contrast conditions. For neutrons, this can be accomplished by isotopic substitution or by using polarized neutrons interacting with magnetic layers. For x-rays, one can vary the x-ray energy to modify the relative scattering contrasts. In any case, one then attempts to simultaneously fit the combined dataset in a self-consistent way; this narrows the range of possible solutions and thus recovers some phase information.

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 The '''Fourier transform''' is a mathematical transformation that appears in many branches of physics. One feature of this transform is that periodic signals in the input stream are converted into well-defined peaks in the output Fourier space.
-The Fourier transform is central in [[scattering]]. In scattering, an incident wave travels through a sample. All of the entities in the sample act as scattering sources, giving rise to secondary waves that interfere with one another. This wave-interference phenomenon is essentially physically performing the Fourier transform operation. Thus, the observed scattering pattern (patterns of constructive and destructive interference) is the Fourier transform of the[[realspace]] density profile probed by the wave. The full 3D [[reciprocal-space]] is the Fourier transform of the sample's structure.
+The Fourier transform is central in [[scattering]]. In scattering, an incident wave travels through a sample. All of the entities in the sample act as scattering sources, giving rise to secondary waves that interfere with one another. This wave-interference phenomenon is essentially physically performing the Fourier transform operation. Thus, the observed scattering pattern (patterns of constructive and destructive interference) is the Fourier transform of the [[realspace]] density profile probed by the wave. The full 3D [[reciprocal-space]] is the Fourier transform of the sample's structure.
 ==Mathematical form==

← Older revision		Revision as of 21:37, 5 June 2014
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	Alternatively, some methods exploit a high degree of coherence in the incident radiation in order to reconstruct the realspace structure (e.g. [[Coherent Diffraction Imaging]]).		Alternatively, some methods exploit a high degree of coherence in the incident radiation in order to reconstruct the realspace structure (e.g. [[Coherent Diffraction Imaging]]).
		+
		+	==See Also==
		+	* [http://en.wikipedia.org/wiki/Fourier_transform Wikipedia: Fourier transform]

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 \end{alignat}
 </math>
-As as be seen, this is mathematically identical to the Fourier transform operation previously described.
+As as be seen, this is mathematically identical to the (three-dimensional) Fourier transform operation previously described.
 ==Phase problem==